Time-Optimal Paths for a Dubins Car and Dubins Airplane with a Unidirectional Turning Constraint.

Abstract

The primary goal of an aircraft emergency landing planner is to safely and efficiently land on a runway. One commonly used tool is the solution} of Dubins problem, which defines minimal-time paths for vehicles moving in a plane with constraints on turn rate. The Dubins solution presumes that vehicles can follow straight paths and turn in both directions. A vehicle, however, can be constrained to unidirectional (i.e., either clockwise or counterclockwise) turning motions after experiencing severe structural damage and/or control failure. A unidirectional turning constraint specifies lower and upper bounds on turn rate, both of the same sign. This dissertation addresses, for the first time, the problem of finding time-optimal paths for Dubins vehicles constrained to unidirectional turning motions.
This dissertation initially considers a Dubins vehicle in a plane, called Dubins car, with unidirectional turning constraints for which the optimal paths are characterized by employing Pontryagin's minimum principle. A geometric interpretation of the identified extremal paths enables direct identification of the optimal path. To extend these planar results to aircraft emergency landing planning, it is necessary to consider the unidirectional Dubins airplane where the planar motions of this unidirectional Dubins car are supplemented by allowing changes in altitude. Optimal paths for the unidirectional Dubins airplane have one of two transition times: the shortest time for the unidirectional Dubins car or the time equal to the absolute altitude difference required to descend divided by the maximum vertical rate. In some instances of the latter case, a suboptimal path must be constructed to guarantee a feasible solution.
Both the unidirectional Dubins car and airplane algorithms developed in this dissertation can be implemented in real-time, thus integrated easily into embedded vehicle management systems. Moreover, these algorithms are complete, thus can be guaranteed to find a feasible solution which in most cases is also time-optimal. Throughout the dissertation, the proposed algorithms are validated through a series of test cases.
The dissertation applies the unidirectional Dubins airplane algorithm to aircraft emergency landing at LaGuardia airport to demonstrate its ability to rapidly identify and present a full suite of landing options to the pilot and automation